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JEE Mathematics Solved Question Paper 2015

Multiple Choice Questions

11.

The shortest distance between the lines $\frac{x - 7}{3} = \frac{y + 4}{- 16} = \frac{z - 6}{7}$ and $\frac{x - 10}{3} = \frac{y - 30}{8} = \frac{4 - z}{5}$ is

$\frac{234}{7}$ units
$\frac{288}{21}$ units
$\frac{221}{3}$ units
$\frac{234}{21}$ units

12.
At t = 0, the function f(t) = $\frac{\sin (t)}{t}$ has

a minimum

a discontinuity

a point of inflexion

a maximum

a maximum

$Given, f (t) = \frac{\sin (t)}{t}$

At t = 0, first we will check continuity of the function

Now, LHL = f(0 - h)

$= \lim_{h \to 0} \frac{\sin (0 - h)}{(0 - h)} = \lim_{h \to 0} \frac{- \sin (h)}{- h} = 1$

$RHL = f (0 + h) = \lim_{h \to 0} \frac{\sin (0 + h)}{(0 + h)} = \lim_{h \to 0} \frac{\sin (h)}{h} = 1$

and f(0) = 1

Since, LHL = RHL = f(0)

So, the function is continuous att = 0.

Now, we check the function is maximum or minimum

$f' (t) = \frac{1}{t} \cos (t) - \frac{1}{t^{2}} \sin (t) and f'' (t) = \frac{- 1}{t} \sin (t) - \frac{1}{t^{2}} \cos (t) - \frac{1}{t^{2}} \cos (t) + \frac{2}{t^{3}} \sin (t) = \frac{- \sin (t)}{t} - \frac{2 \cos (t)}{t^{2}} + \frac{2 \sin (t)}{t^{3}}$

For maximum or minimum value of f(x), put

$f' (x) = 0 \Rightarrow \frac{\cos (t)}{t} - \frac{\sin (t)}{t^{2}} = 0 \Rightarrow \frac{\tan (t)}{t} = 1$

$Now, \lim_{t \to 0} f'' (t) = - \lim_{t \to 0} (\frac{\sin (t)}{t}) - 2 \lim_{t \to 0} (\frac{tcos (t) - \sin (t)}{t^{3}}) [\frac{0}{0} form] = - 1 - 2 \lim_{t \to 0} (\frac{\cos (t) - tsin (t) - \cos (t)}{3 t^{2}}) [∵ using L' Hospital rule] = - 1 + \frac{2}{3} \lim_{t \to 0} \frac{\sin (t)}{t} = - 1 + \frac{2}{3} \times 1 = \frac{- 1}{3} < 0$

So, function f(t) is maximum at t = 0.

13.

Which of the following inequality is true for x > 0?

$\log (1 + x) < \frac{x}{1 + x} < x$
$\frac{x}{1 + x} < x < \log (1 + x)$
$x < \log (1 + x) < \frac{x}{1 + x}$
$\frac{x}{1 + x} < \log (1 + x) < x$

14.

If the complex number z lies on a circle with centre at the origin and radius = $\frac{1}{4}$ , then the 4 complex number - 1 + 8z lies on a circle with radius

15.

If line y = 2x + c is a normal to the ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{16} = 1$ ,then

c = $\frac{2}{3}$
$\sqrt{\frac{73}{5}}$
$c = \frac{14}{\sqrt{73}}$
$\sqrt{\frac{5}{7}}$

16.

If x² + x + 1 = 0, then the value of $\sum_{n = 1}^{6} {(x^{n} + \frac{1}{x^{n}})}^{2}$ is

17.

If p : It rains today, q : I go to school, r : I shall meet any friends ands : I shall go for a movie, then which of the following is the proportion? If it does not rain or if I do not go to school, then I shall meet my friend and go for a movie.

$(~ p \land ~ q) \Rightarrow (r \land s)$
$~ (p \land q) \Rightarrow (r \land s)$
$~ (p \lor q) \Rightarrow (r \land s)$
None of these

18.

If the matrix A = $[\begin{matrix} 1 & 3 & 1 \\ - 1 & 2 & - 3 \\ 0 & 1 & 2 \end{matrix}]$ , then adj (adj A) is equal to

$[\begin{matrix} 12 & 36 & 12 \\ - 12 & 24 & - 36 \\ 0 & 12 & 24 \end{matrix}]$
$[\begin{matrix} 12 & 26 & - 12 \\ 24 & 36 & - 36 \\ 0 & 12 & - 24 \end{matrix}]$
$[\begin{matrix} 12 & - 12 & 36 \\ 24 & - 24 & - 36 \\ 0 & 12 & 24 \end{matrix}]$
None of the above